### Asymptotics of the Magnitude of Metric Spaces

#### Posted by John Baez

*guest post by Tom Leinster*

Simon Willerton and I have just arXived a new paper, On the asymptotic magnitude of subsets of Euclidean space. It’s about a subject that owes a lot to the $n$-Café: the cardinality of metric spaces. Before we submit it to a journal, we’d be interested and grateful to have your comments — anything from typos to matters of philosophy.

Here’s the idea. Cardinality is supposed to play the same kind of role for metric spaces as ordinary cardinality plays for sets. Now, a fundamental property of the cardinality of finite sets is the inclusion-exclusion principle:

$|A \cup B| = |A| + |B| - |A \cap B|.$

Here $A$ and $B$ are sets and $| |$ means ordinary cardinality. But what if, say, $A$ and $B$ are compact subsets of $\mathbb{R}^n$ and $| |$ means metric cardinality? Does the inclusion-exclusion principle hold?

The answer is ‘no’, but it’s a particularly interesting ‘no’ — probably more interesting than ‘yes’ would have been. It’s ‘no, but asymptotically yes’, at least in the cases we’ve succeeded in analyzing.

To explain this I’ll have to back up a bit.

First, Simon and I decided that ‘the cardinality of a metric space’
was too confusing a phrase: people hearing it for the first time think
it means the cardinality of the set of points. So, we decided to
switch to **magnitude**.

Second, magnitude was only defined for
*finite* metric spaces. But there’s a rather hands-on strategy
for extending the definition to compact metric spaces $A$: take a
sequence $A_0 \subseteq A_1 \subseteq \cdots$ of finite subspaces, with
$\bigcup A_n$ dense in $A$, and try to define

$|A| = \lim_{n \to \infty} |A_n|.$

There are all sorts of things that could go wrong with this strategy, but sometimes it works, or at least works well enough to enable us to get somewhere. For example, we use it to calculate the magnitudes of straight line segments, circles, and middle-third Cantor sets.

Third, a lot is known about ‘measures’ satisfying the
inclusion-exclusion principle on subsets of $\mathbb{R}^n$. These are known as
**valuations**. If you restrict to **polyconvex** sets
(finite unions of compact convex sets) then the
valuations are completely classified. This is Hadwiger’s
Theorem. (I’m glossing over some details here: really we want the
valuations to have a couple of further properties. See the
link.) For example, Hadwiger tells us
that any valuation on $\mathbb{R}^2$ must be of the form

$\alpha_0 \cdot Euler characteristic + \alpha_1 \cdot perimeter + \alpha_2 \cdot area$

where $\alpha_0$, $\alpha_1$ and $\alpha_2$ are constants. We’ll want to go outside the world of polyconvex sets, since there are many common non-polyconvex spaces for which Euler characteristic, perimeter etc. have obvious meanings (e.g. circles). But we’ll continue to use Hadwiger’s Theorem as a guide.

Finally, there’s a pesky technical point that I feel obliged to put in for anyone taking the trouble to follow in detail. Skip it if you’re not. The original definition of magnitude/cardinality involved some terms of the form $e^{-2d(a,b)}$. We switched to $e^{-d(a,b)}$. So if, in what follows, you’re ever surprised by a factor of $2$, you know why.

Now I can tell the story! We’ve known for a while that the magnitude of a line segment (closed interval) of length $L$ is

$1 + L/2 = 1 \cdot Euler characteristic + \frac{1}{2} \cdot perimeter.$

Let’s consider the magnitude of, say, ‘nice’ subsets of
$\mathbb{R}^2$. *If* magnitude satisfies the
inclusion-exclusion principle, then Hadwiger’s Theorem suggests that
it is probably a linear combination of Euler characteristic, perimeter
and area. The example of the line segment tells us what the first two
coefficients must be. So, if circles are ‘nice’ then the magnitude of
a circle $C_L$ of circumference $L$ should be

$1 \cdot Euler characteristic + \frac{1}{2} \cdot perimeter = 1 \cdot 0 + \frac{1}{2} \cdot L = L/2.$

Is that true? No! But it’s *asymptotically* true; that is,

$\lim_{L \to \infty} (|C_L| - L/2) = 0.$

Crudely put: for large $L$,

$|C_L| \approx 1 \cdot Euler characteristic + \frac{1}{2} \cdot perimeter.$

There’s an infinite family of sensible metrics you can put on the circle, and this result holds for all of them. The proof — even for the plain old subspace metric — uses some techniques of asymptotic analysis. Thanks to Bruce Bartlett for tracking down the key result.

Now let’s try another example: the Cantor set, defined as usual by successively removing middle thirds from a line segment. Let $T_L$ be the Cantor set whose endpoints are distance $L$ apart. This is a different kind of animal from the line or the circle: it’s not obvious what Euler characteristic and perimeter would even mean. (On the other hand, we do expect a nonzero measure of dimension $\log_3 2$, since that’s the Hausdorff dimension of $T_L$.) Let’s use the self-similarity of the Cantor set instead. Since $T_L$ consists of two disjoint copies of $T_{L/3}$, the inclusion-exclusion principle suggests

$|T_L| = 2|T_{L/3}|.$

Is that true? No! But it’s *asymptotically* true; that is,

$\lim_{L \to \infty}(|T_L| - p(L)) = 0$

for a certain function $p$ satisfying

$p(L) = 2p(L/3).$

(It follows that $|T_L|$ grows like $L^{\log_3 2}$.) Crudely put: for large $L$,

$|T_L| \approx 2|T_{L/3}|.$

Here’s an intuitive explanation. Magnitude respects coproducts,
meaning that if $A$ and $B$ are metric spaces and we write $A + B$ for
their ‘distant union’ (the disjoint union with $d(a, b) = \infty$ for
all $a \in A$ and $b \in B$) then $|A + B| = |A| + |B|$. The union
$T_L = T_{L/3} \cup T_{L/3}$ is disjoint. It’s not distant, since the
two copies of $T_{L/3}$ are distance $L/3$, not $\infty$, apart —
but when $L$ is large, it’s *nearly* distant, so the magnitude
formula nearly holds.

(Incidentally, there’s a little surprise: it turns out that inside the Cantor set, a periodic function is hiding! But you’ll have to read the paper to find out about that.)

All this evidence led us to make a conjecture. Roughly, it
says that the inclusion-exclusion principle is satisfied
*asymptotically* for *compact* subsets of $\mathbb{R}^n$,
and *exactly* for *convex* sets.

Strong Asymptotic ConjectureThere is a unique function $P: \{ compact subsets of \mathbb{R}^n \} \to \mathbb{R}$ such that

- $\lim_{t \to \infty} (|t A| - P(t A)) = 0$ for all $A$, where $t A$ denotes $A$ scaled up by a factor of $t$
- $P$ satisfies the inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and $P(\emptyset) = 0$
- when $A$ is polyconvex, $P(A)$ is a certain specific linear combination of the intrinsic volumes of $A$
- when $A$ is convex, $|A| = P(A)$ (i.e. $|A|$ is
exactlythat linear combination of intrinsic volumes).

Atiyah is supposed to have said something like ‘if you don’t understand a conjecture, generalize it’. Our conjecture might be too strong — and we have a Weak Asymptotic Conjecture too — but we believe that something like it is true.

## Re: Asymptotics of the Magnitude of Metric Spaces

Maybe I should read your paper first, but I would have said that that's $1/4$ of the perimeter. Otherwise you'll be claiming that the limit of the magnitude of a very thin rectangle as its width (and hence area) goes to zero is almost twice the magnitude of a line segment: $\lim_{W \to 0} \left(1 \cdot 1 + \frac{1}{2} \cdot (2L + 2W) + \alpha_2 \cdot LW\right) = 1 + L \ne 1 + L/2 ,$ which I find hard to believe if this is supposed to measure compact

subsetsof $\mathbb{R}^2$.